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Properties

Label 50.12.d.b.21.1
Level $50$
Weight $12$
Character 50.21
Analytic conductor $38.417$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,12,Mod(11,50)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("50.11"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.4171590280\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 21.1
Character \(\chi\) \(=\) 50.21
Dual form 50.12.d.b.31.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(9.88854 - 30.4338i) q^{2} +(-619.017 - 449.742i) q^{3} +(-828.433 - 601.892i) q^{4} +(-1807.38 + 6749.93i) q^{5} +(-19808.6 + 14391.8i) q^{6} -82655.9 q^{7} +(-26509.9 + 19260.5i) q^{8} +(126173. + 388320. i) q^{9} +(187554. + 121752. i) q^{10} +(-99869.4 + 307366. i) q^{11} +(242118. + 745163. i) q^{12} +(-444325. - 1.36749e6i) q^{13} +(-817347. + 2.51553e6i) q^{14} +(4.15453e6 - 3.36546e6i) q^{15} +(324028. + 997255. i) q^{16} +(854128. - 620560. i) q^{17} +1.30657e7 q^{18} +(-1.35459e7 + 9.84170e6i) q^{19} +(5.56002e6 - 4.50402e6i) q^{20} +(5.11654e7 + 3.71739e7i) q^{21} +(8.36677e6 + 6.07881e6i) q^{22} +(-8.05235e6 + 2.47826e7i) q^{23} +2.50724e7 q^{24} +(-4.22949e7 - 2.43994e7i) q^{25} -4.60117e7 q^{26} +(5.46555e7 - 1.68212e8i) q^{27} +(6.84749e7 + 4.97499e7i) q^{28} +(-1.10638e8 - 8.03832e7i) q^{29} +(-6.13417e7 - 1.59718e8i) q^{30} +(-1.73361e8 + 1.25954e8i) q^{31} +3.35544e7 q^{32} +(2.00057e8 - 1.45350e8i) q^{33} +(-1.04399e7 - 3.21308e7i) q^{34} +(1.49391e8 - 5.57921e8i) q^{35} +(1.29201e8 - 3.97640e8i) q^{36} +(1.48199e7 + 4.56111e7i) q^{37} +(1.65571e8 + 5.09574e8i) q^{38} +(-3.39974e8 + 1.04633e9i) q^{39} +(-8.20938e7 - 2.13751e8i) q^{40} +(-1.96165e8 - 6.03734e8i) q^{41} +(1.63729e9 - 1.18956e9i) q^{42} -6.54088e8 q^{43} +(2.67737e8 - 1.94522e8i) q^{44} +(-2.84917e9 + 1.49815e8i) q^{45} +(6.74602e8 + 4.90127e8i) q^{46} +(-1.81653e9 - 1.31979e9i) q^{47} +(2.47929e8 - 7.63047e8i) q^{48} +4.85467e9 q^{49} +(-1.16080e9 + 1.04592e9i) q^{50} -8.07812e8 q^{51} +(-4.54988e8 + 1.40031e9i) q^{52} +(9.62845e8 + 6.99548e8i) q^{53} +(-4.57888e9 - 3.32675e9i) q^{54} +(-1.89420e9 - 1.22964e9i) q^{55} +(2.19120e9 - 1.59200e9i) q^{56} +1.28114e10 q^{57} +(-3.54042e9 + 2.57226e9i) q^{58} +(-1.42880e9 - 4.39739e9i) q^{59} +(-5.46740e9 + 2.87486e8i) q^{60} +(-1.80670e9 + 5.56044e9i) q^{61} +(2.11897e9 + 6.52153e9i) q^{62} +(-1.04289e10 - 3.20969e10i) q^{63} +(3.31804e8 - 1.02119e9i) q^{64} +(1.00335e10 - 5.27581e8i) q^{65} +(-2.44527e9 - 7.52578e9i) q^{66} +(-8.15756e9 + 5.92681e9i) q^{67} -1.08110e9 q^{68} +(1.61303e10 - 1.17194e10i) q^{69} +(-1.55024e10 - 1.00636e10i) q^{70} +(1.26835e10 + 9.21511e9i) q^{71} +(-1.08241e10 - 7.86415e9i) q^{72} +(6.35149e9 - 1.95479e10i) q^{73} +1.53467e9 q^{74} +(1.52078e10 + 3.41254e10i) q^{75} +1.71455e10 q^{76} +(8.25480e9 - 2.54057e10i) q^{77} +(2.84820e10 + 2.06934e10i) q^{78} +(-2.04851e10 - 1.48833e10i) q^{79} +(-7.31704e9 + 3.84743e8i) q^{80} +(-5.09690e10 + 3.70311e10i) q^{81} -2.03137e10 q^{82} +(3.52620e10 - 2.56194e10i) q^{83} +(-2.00125e10 - 6.15922e10i) q^{84} +(2.64500e9 + 6.88689e9i) q^{85} +(-6.46797e9 + 1.99064e10i) q^{86} +(3.23351e10 + 9.95172e10i) q^{87} +(-3.27252e9 - 1.00718e10i) q^{88} +(-1.04280e10 + 3.20942e10i) q^{89} +(-2.36147e10 + 8.81926e10i) q^{90} +(3.67261e10 + 1.13031e11i) q^{91} +(2.15873e10 - 1.56841e10i) q^{92} +1.63960e11 q^{93} +(-5.81289e10 + 4.22331e10i) q^{94} +(-4.19481e10 - 1.09222e11i) q^{95} +(-2.07708e10 - 1.50909e10i) q^{96} +(9.74175e10 + 7.07779e10i) q^{97} +(4.80056e10 - 1.47746e11i) q^{98} -1.31957e11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 448 q^{2} - 263 q^{3} - 14336 q^{4} + 1770 q^{5} - 8416 q^{6} - 111844 q^{7} - 458752 q^{8} - 1174523 q^{9} + 304960 q^{10} + 207277 q^{11} + 1026048 q^{12} + 893677 q^{13} - 1270048 q^{14} + 4696640 q^{15}+ \cdots - 505737997606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(e\left(\frac{3}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.88854 − 30.4338i 0.218508 − 0.672499i
\(3\) −619.017 − 449.742i −1.47074 − 1.06856i −0.980398 − 0.197025i \(-0.936872\pi\)
−0.490342 − 0.871530i \(-0.663128\pi\)
\(4\) −828.433 − 601.892i −0.404508 − 0.293893i
\(5\) −1807.38 + 6749.93i −0.258651 + 0.965971i
\(6\) −19808.6 + 14391.8i −1.03997 + 0.755583i
\(7\) −82655.9 −1.85881 −0.929404 − 0.369063i \(-0.879679\pi\)
−0.929404 + 0.369063i \(0.879679\pi\)
\(8\) −26509.9 + 19260.5i −0.286031 + 0.207813i
\(9\) 126173. + 388320.i 0.712249 + 2.19208i
\(10\) 187554. + 121752.i 0.593096 + 0.385015i
\(11\) −99869.4 + 307366.i −0.186970 + 0.575436i −0.999977 − 0.00681917i \(-0.997829\pi\)
0.813006 + 0.582255i \(0.197829\pi\)
\(12\) 242118. + 745163.i 0.280886 + 0.864479i
\(13\) −444325. − 1.36749e6i −0.331904 − 1.02149i −0.968227 − 0.250072i \(-0.919546\pi\)
0.636324 − 0.771422i \(-0.280454\pi\)
\(14\) −817347. + 2.51553e6i −0.406165 + 1.25005i
\(15\) 4.15453e6 − 3.36546e6i 1.41260 − 1.14431i
\(16\) 324028. + 997255.i 0.0772542 + 0.237764i
\(17\) 854128. − 620560.i 0.145900 − 0.106002i −0.512441 − 0.858722i \(-0.671259\pi\)
0.658341 + 0.752720i \(0.271259\pi\)
\(18\) 1.30657e7 1.62980
\(19\) −1.35459e7 + 9.84170e6i −1.25506 + 0.911854i −0.998504 − 0.0546759i \(-0.982587\pi\)
−0.256555 + 0.966530i \(0.582587\pi\)
\(20\) 5.56002e6 − 4.50402e6i 0.388518 − 0.314728i
\(21\) 5.11654e7 + 3.71739e7i 2.73382 + 1.98624i
\(22\) 8.36677e6 + 6.07881e6i 0.346125 + 0.251475i
\(23\) −8.05235e6 + 2.47826e7i −0.260867 + 0.802866i 0.731750 + 0.681574i \(0.238704\pi\)
−0.992617 + 0.121293i \(0.961296\pi\)
\(24\) 2.50724e7 0.642737
\(25\) −4.22949e7 − 2.43994e7i −0.866199 − 0.499699i
\(26\) −4.60117e7 −0.759477
\(27\) 5.46555e7 − 1.68212e8i 0.733050 − 2.25609i
\(28\) 6.84749e7 + 4.97499e7i 0.751904 + 0.546290i
\(29\) −1.10638e8 − 8.03832e7i −1.00165 − 0.727740i −0.0392076 − 0.999231i \(-0.512483\pi\)
−0.962441 + 0.271491i \(0.912483\pi\)
\(30\) −6.13417e7 − 1.59718e8i −0.460881 − 1.20001i
\(31\) −1.73361e8 + 1.25954e8i −1.08758 + 0.790174i −0.978989 − 0.203912i \(-0.934634\pi\)
−0.108592 + 0.994086i \(0.534634\pi\)
\(32\) 3.35544e7 0.176777
\(33\) 2.00057e8 − 1.45350e8i 0.889869 − 0.646528i
\(34\) −1.04399e7 − 3.21308e7i −0.0394061 − 0.121280i
\(35\) 1.49391e8 − 5.57921e8i 0.480783 − 1.79555i
\(36\) 1.29201e8 − 3.97640e8i 0.356125 − 1.09604i
\(37\) 1.48199e7 + 4.56111e7i 0.0351348 + 0.108134i 0.967086 − 0.254450i \(-0.0818946\pi\)
−0.931951 + 0.362584i \(0.881895\pi\)
\(38\) 1.65571e8 + 5.09574e8i 0.338980 + 1.04327i
\(39\) −3.39974e8 + 1.04633e9i −0.603379 + 1.85701i
\(40\) −8.20938e7 − 2.13751e8i −0.126759 − 0.330049i
\(41\) −1.96165e8 − 6.03734e8i −0.264430 − 0.813831i −0.991824 − 0.127611i \(-0.959269\pi\)
0.727395 − 0.686219i \(-0.240731\pi\)
\(42\) 1.63729e9 − 1.18956e9i 1.93311 − 1.40448i
\(43\) −6.54088e8 −0.678515 −0.339257 − 0.940694i \(-0.610176\pi\)
−0.339257 + 0.940694i \(0.610176\pi\)
\(44\) 2.67737e8 − 1.94522e8i 0.244747 − 0.177819i
\(45\) −2.84917e9 + 1.49815e8i −2.30171 + 0.121028i
\(46\) 6.74602e8 + 4.90127e8i 0.482925 + 0.350865i
\(47\) −1.81653e9 − 1.31979e9i −1.15532 − 0.839393i −0.166145 − 0.986101i \(-0.553132\pi\)
−0.989180 + 0.146709i \(0.953132\pi\)
\(48\) 2.47929e8 − 7.63047e8i 0.140443 − 0.432240i
\(49\) 4.85467e9 2.45517
\(50\) −1.16080e9 + 1.04592e9i −0.525318 + 0.473329i
\(51\) −8.07812e8 −0.327850
\(52\) −4.54988e8 + 1.40031e9i −0.165952 + 0.510747i
\(53\) 9.62845e8 + 6.99548e8i 0.316256 + 0.229774i 0.734576 − 0.678526i \(-0.237381\pi\)
−0.418320 + 0.908300i \(0.637381\pi\)
\(54\) −4.57888e9 − 3.32675e9i −1.35704 − 0.985950i
\(55\) −1.89420e9 − 1.22964e9i −0.507494 − 0.329445i
\(56\) 2.19120e9 − 1.59200e9i 0.531676 − 0.386285i
\(57\) 1.28114e10 2.82023
\(58\) −3.54042e9 + 2.57226e9i −0.708273 + 0.514590i
\(59\) −1.42880e9 − 4.39739e9i −0.260187 − 0.800772i −0.992763 − 0.120088i \(-0.961682\pi\)
0.732577 − 0.680685i \(-0.238318\pi\)
\(60\) −5.46740e9 + 2.87486e8i −0.907713 + 0.0477292i
\(61\) −1.80670e9 + 5.56044e9i −0.273887 + 0.842936i 0.715625 + 0.698484i \(0.246142\pi\)
−0.989512 + 0.144452i \(0.953858\pi\)
\(62\) 2.11897e9 + 6.52153e9i 0.293746 + 0.904056i
\(63\) −1.04289e10 − 3.20969e10i −1.32393 − 4.07465i
\(64\) 3.31804e8 − 1.02119e9i 0.0386271 − 0.118882i
\(65\) 1.00335e10 − 5.27581e8i 1.07258 − 0.0563982i
\(66\) −2.44527e9 − 7.52578e9i −0.240346 − 0.739707i
\(67\) −8.15756e9 + 5.92681e9i −0.738157 + 0.536302i −0.892133 − 0.451772i \(-0.850792\pi\)
0.153977 + 0.988075i \(0.450792\pi\)
\(68\) −1.08110e9 −0.0901709
\(69\) 1.61303e10 − 1.17194e10i 1.24157 − 0.902057i
\(70\) −1.55024e10 − 1.00636e10i −1.10245 − 0.715669i
\(71\) 1.26835e10 + 9.21511e9i 0.834292 + 0.606149i 0.920770 − 0.390105i \(-0.127561\pi\)
−0.0864780 + 0.996254i \(0.527561\pi\)
\(72\) −1.08241e10 − 7.86415e9i −0.659268 − 0.478986i
\(73\) 6.35149e9 − 1.95479e10i 0.358591 − 1.10363i −0.595307 − 0.803499i \(-0.702969\pi\)
0.953898 − 0.300132i \(-0.0970306\pi\)
\(74\) 1.53467e9 0.0803970
\(75\) 1.52078e10 + 3.41254e10i 0.739997 + 1.66051i
\(76\) 1.71455e10 0.775669
\(77\) 8.25480e9 − 2.54057e10i 0.347542 − 1.06962i
\(78\) 2.84820e10 + 2.06934e10i 1.11699 + 0.811543i
\(79\) −2.04851e10 − 1.48833e10i −0.749013 − 0.544190i 0.146508 − 0.989210i \(-0.453197\pi\)
−0.895521 + 0.445020i \(0.853197\pi\)
\(80\) −7.31704e9 + 3.84743e8i −0.249655 + 0.0131273i
\(81\) −5.09690e10 + 3.70311e10i −1.62419 + 1.18005i
\(82\) −2.03137e10 −0.605080
\(83\) 3.52620e10 − 2.56194e10i 0.982602 − 0.713902i 0.0243135 − 0.999704i \(-0.492260\pi\)
0.958289 + 0.285802i \(0.0922600\pi\)
\(84\) −2.00125e10 − 6.15922e10i −0.522114 − 1.60690i
\(85\) 2.64500e9 + 6.88689e9i 0.0646579 + 0.168352i
\(86\) −6.46797e9 + 1.99064e10i −0.148261 + 0.456300i
\(87\) 3.23351e10 + 9.95172e10i 0.695534 + 2.14063i
\(88\) −3.27252e9 − 1.00718e10i −0.0661040 − 0.203447i
\(89\) −1.04280e10 + 3.20942e10i −0.197951 + 0.609231i 0.801978 + 0.597353i \(0.203781\pi\)
−0.999929 + 0.0118778i \(0.996219\pi\)
\(90\) −2.36147e10 + 8.81926e10i −0.421550 + 1.57434i
\(91\) 3.67261e10 + 1.13031e11i 0.616945 + 1.89876i
\(92\) 2.15873e10 − 1.56841e10i 0.341479 − 0.248099i
\(93\) 1.63960e11 2.44389
\(94\) −5.81289e10 + 4.22331e10i −0.816938 + 0.593540i
\(95\) −4.19481e10 − 1.09222e11i −0.556201 − 1.44820i
\(96\) −2.07708e10 − 1.50909e10i −0.259993 − 0.188896i
\(97\) 9.74175e10 + 7.07779e10i 1.15184 + 0.836861i 0.988725 − 0.149746i \(-0.0478455\pi\)
0.163116 + 0.986607i \(0.447845\pi\)
\(98\) 4.80056e10 − 1.47746e11i 0.536474 − 1.65110i
\(99\) −1.31957e11 −1.39457
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 50.12.d.b.21.1 ✓ 56
25.6 even 5 inner 50.12.d.b.31.1 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.12.d.b.21.1 ✓ 56 1.1 even 1 trivial
50.12.d.b.31.1 yes 56 25.6 even 5 inner